Character mistake: The finger bone found in the dead man's hand was said to come from somewhere near Assateague Island, to which Hodgins replies, "That's where the money pit is." The "Money Pit" is actually located on Oak Island in Nova Scotia, not on Assateague Island, in Maryland/Virginia. He also says that they dug it out to about 150 feet, which is also incorrect. The 1971 Triton excavation was able to dig a shaft to a depth of 235 feet. What is said to be on Assateague island is the legendary lost Spanish galleon La Galga, not the Money Pit.
Character mistake: When Booth is locked in the ship, his "hallucination" asks Booth why he never bothered to learn some lock picking skills. However, in the episode "Judas on a Pole," Bones tells Booth that she's been practicing some lock picking skills that Booth taught her.
Character mistake: When Bones and Dr Hodgins, who is proclaimed as a seasoned diver in a previous episode, are in the tank, Bones gives an instruction and Hodgins replies with a thumbs up. Since Bones is diving as well, they both would know from certification (without which neither would be allowed to dive in the facility) classes that a thumbs up is the hand signal for "I'm ascending", not OK or any form of acknowledgement. They either are not certified divers or they should have immediately surfaced.
Character mistake: When Booth mentions the 2007 Led Zeppelin concert in London, he comments that they toured that year. That Zeppelin concert was not part of a tour, but just a one-off reunion to honor the memory of late Atlantic Records executive Ahmet Ertegun.
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Character mistake: Zach, from his hospital bed, says he used 3 grams of the monomer to make the explosion. 3 grams of a powder would be a very small amount; less than a teaspoonful. But when he actually did use it, he took a loose amount in a large tablespoon. Zach would have carefully measured it in order to be precise; but he did not. So this is a factual error in what he says; it is also a character mistake in that Zach is the personification of technical accuracy, and would never take a casual spoonful - especially of an explosive.
Character mistake: In season 2, episode 4, when they find the girl buried under the cross, you can see her face, which means she was on her back, because her neck couldn't possibly bend that far around without it being noted by Dr. Brennan later in the episode (which it isn't). Later in the episode, however, Special Agent Booth tells Epps' wife that she was found buried face down.